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\section{The Macro Model}

\subsection{Production Technology}

We model economic activity in continuous time, indexed by $t$. The state variables, the controls, and the technology variables thus are functions of $t$. We shall usually simplify notation, however, by omitting
time as an explicit argument.

There is a
single consumption good in the economy. We assume that the per capita output
of the good can be written as a linear function of a per capita stock of
capital $k$:\footnote{%
We assume that technological progress allows the \textquotedblleft
productive services" supplied by inputs to expand even if the physical
inputs stop growing. In particular, we implicitly assume that labor input
can be expanded through investment in human capital even if hours and number
of employees remain fixed. Hence, the marginal product of capital does not
decline as $k$ accumulates.} 
\begin{equation}
y=Ak  \label{eq:ProdFn}
\end{equation}%
We assume that capital depreciates at the rate $\delta $, while investment
in new capital is denoted by $i$: 
\begin{equation}
\dot{k}=i-\delta k  \label{eq:kdot}
\end{equation}%
Energy is also needed to produce output. At each instant, the ratio of
energy to capital inputs is fixed. Denote the per capita energy derived from
fossil fuel resources that is used to produce goods by $R\geq 0$. Per capita renewable energy supplied by the backstop technology $B\geq 0 $ is a perfect substitute for the energy produced from fossil fuel burning. Thus, we assume that at each moment of
time: 
\begin{equation}
R+B=y  \label{eq:NrgInput}
\end{equation}%

Letting $c$ denote per capita consumption, we assume that the lifetime
utility function is given by: 
\begin{equation}
U=\max \int_{0}^{\infty }e^{-\beta \tau }\frac{c(\tau )^{1-\gamma }}{%
1-\gamma }\,d\tau  \label{eq:Objective}
\end{equation}

The term $e^{-\beta \tau }$ acts as a discount factor, capturing the fact
that utility from future consumption is less valuable than today's
consumption.

\subsection{Fossil Fuel Supply}

Let $Q$ denote
the (exogenous) population and labor supply and assume that it grows at the
constant rate $\pi $. The total fossil fuel used will then be $QR$.
Following Heal (1976) and Solow and Wan (1976), we assume that the most
easily-mined, or the richest deposits or fields, tend to be exhausted first.
The marginal resource costs of extraction then increase with the total
quantity of resources mined to date, $S$, which is also the integral of $QR$%
: 
\begin{equation}
\dot{S}=QR  \label{eq:Sdot}
\end{equation}%
Heal introduced the idea of an increasing marginal cost of extraction to
show that the optimal price of an exhaustible resource begins above marginal
cost, and falls toward it over time.\footnote{%
This claim is rigorously proved in Oren and Powell (1985).}

We modify the resource depletion model to also allow for technical change in
mining exploration. The marginal cost of extraction, $g(S,N)$, depends not
only on $S$ but also the state of technical knowledge $N$. It is useful to define the energy supplies in efficiency units. Improvements in energy efficiency then also will lead to reduction in the
per-unit cost $g(S,N)$ of supplying an additional unit of $R$. Investment in
mining technology, or the efficiency with which fossil fuel is used to
provide useful energy services, leads to an accumulation of knowledge: 
\begin{equation}
\dot{N}=n  \label{eq:Ndot}
\end{equation}%
We then assume that $g(S,N)$ is given by the following function: 
\begin{equation}
g(S,N)=\alpha _{0}+\frac{\alpha _{1}}{\bar{S}-S-\alpha _{2}/(\alpha
_{3}+N)}=\alpha _{0}+\frac{\alpha _{1}(\alpha _{3}+N)}{(\bar{S}-S)(\alpha
_{3}+N)-\alpha _{2}}  \label{eq:MCMining}
\end{equation}%
illustrated in Figure \ref{fig:MiningTC}. For a given state of technical
knowledge $N$, the maximum fossil fuel resource that can be extracted is
given by $\bar{S}-\alpha _{2}/(\alpha _{3}+N)$. The terms $\alpha
_{0},\alpha _{1},\alpha _{2}$ and $\alpha _{3}$ in \eqref{eq:MCMining} are
parameters.

\begin{figure}[ht]
\centering \includegraphics[width=6in]{MiningTC.pdf}
\caption{Cost of energy from fossil fuels}
\label{fig:MiningTC}
\end{figure}

The absolute maximum fossil fuel available is given by $\bar{S}$, and this
is only available asymptotically as the stock of investment in new fossil
fuel technology $N\to\infty$. Even then, to exploit all the technically
available resources $\bar{S}$, would incur arbitrarily large costs.

For the later analysis, it also is useful to derive the partial derivatives
of the fossil fuel cost function $g(S,N)$. The fist partial derivatives are
given by 
\begin{equation}
\frac{\partial g}{\partial S}=\frac{\alpha_1(\alpha_3+N)^2}{[(\bar{S}%
-S)(\alpha_3+N)-\alpha_2]^2}>0  \label{eq:PartialgPartialS}
\end{equation}
and 
\begin{equation}
\frac{\partial g}{\partial N}=-\frac{\alpha_1\alpha_2}{[(\bar{S}%
-S)(\alpha_3+N)-\alpha_2]^2}<0  \label{eq:PartialgPartialN}
\end{equation}
so that increases in $S$ increase marginal cost, while improved technology
reduces the costs of providing fossil fuel energy. The second order partial
derivatives with respect to $S$ and $N$ are given by 
\begin{equation}
\frac{\partial^2 g}{\partial S^2}=\frac{2\alpha_1(\alpha_3+N)^3}{[(\bar{S}%
-S)(\alpha_3+N)-\alpha_2]^3}>0  \label{eq:Partial2gPartialS2}
\end{equation}
and 
\begin{equation}
\frac{\partial^2 g}{\partial N^2}=\frac{2\alpha_1\alpha_2(\bar{S}-S)}{[(\bar{%
S}-S)(\alpha_3+N)-\alpha_2]^3}>0  \label{eq:Partial2gPartialN2}
\end{equation}
In particular, this function implies that cumulative exploitation $S$
increases fossil fuel energy cost at an increasing rate, while investment in
fossil fuel technology decreases costs at a decreasing rate. In fact, we can
conclude from \eqref{eq:PartialgPartialN} that $\partial g/\partial
N\rightarrow 0$ as $N\rightarrow\infty$. The latter fact should imply that
eventually it becomes uneconomic to invest further in reducing the costs of
fossil fuel energy. Thus, fossil fuel resources will likely be abandoned
long before all known deposits are exhausted as rising costs make renewable
energy technologies more attractive.

Finally, the cross second partial derivative will be given by 
\begin{equation}
\frac{\partial^2 g}{\partial N\partial S}=-\frac{2\alpha_1\alpha_2(%
\alpha_3+N)}{[(\bar{S}-S)(\alpha_3+N)-\alpha_2]^3}<0
\label{eq:Partial2gPartialSN}
\end{equation}
Hence, investment in fossil fuel technology delays the increase in costs of
fossil fuel energy accompanying increased exploitation.

For energy to be productive on net, we need the value of output produced
from energy input to exceed the costs of producing that energy input. In
particular, if only fossil fuel is used to provide energy input, we must
have $1>g(S,N)$. The function \eqref{eq:MCMining} assumed above implies that
exhaustion of fossil fuel resources must eventually increase costs $g(S,N)$
so that this constraint is violated.

\subsection{Backstop Renewable Energy Technologies}

Motivated by the analysis that uses learning curves, we assume that the
marginal cost $p$ (measured in terms of goods) of the energy services
produced using the backstop technology declines as new knowledge is gained.
Following the literature examining learning-by-doing, we assume that
experience constructing capital using renewable energy input is the primary
factor in lowering the amount of such capital required to harvest the energy
needed to produce a given level of output. Even so, there is a limit, $%
\Gamma _{2}$, determined by physical constraints, below which $p$ cannot
fall. Explicitly, using $H$ to denote the stock of knowledge about backstop
energy production, and $\Gamma _{1}$ the initial value of $p$ (when $H=0$),
we assume: 
\begin{equation}
p=%
\begin{cases}
(\Gamma _{1}+H)^{-\alpha } & \text{ if $H\leq \Gamma _{2}{}^{-1/\alpha }-$}%
\Gamma _{1}, \\ 
\Gamma _{2} & \text{otherwise}%
\end{cases}
\label{eq:RenewCost}
\end{equation}%
for constant parameters $\Gamma _{1}$, $\Gamma _{2}$ and $\alpha $. We
assume that $\Gamma _{1}{}^{-\alpha }>g(0,0)$, so that renewable energy is
initially noncompetitive with fossil fuels.

We allow for technological progress to reduce the cost of renewable energy
through a learning curve. In our formulation, some direct R\&D expenditure $j$ can accelerate the accumulation of knowledge about the renewable technology:\footnote{%
Klaassen et. al. (2005) studied the impact of public R\&D and capacity
expansion on cost reducing innovation for wind turbine farms in Denmark,
Germany and the UK. They estimated a two-factor learning curve model that
allowed for both learning-by-doing and direct R\&D. They derive robust
estimates suggesting that direct R\&D is roughly twice as productive for
reducing costs as is learning-by-doing. They interpret their results as
enhancing the validity of the two-factor learning curve formulation.
Kouvaritakis et al. (2000) used a two-factor learning specification that
incorporates learning-by-doing effects as well as a relationship between
technology performance and R\&D expenditure.} 
\begin{equation}
\dot{H}=%
\begin{cases}
B(1+\psi j) & \text{ if $H\leq \Gamma _{2}^{-1/\alpha }-$}\Gamma _{1}, \\ 
0 & \text{otherwise}%
\end{cases}
\label{eq:Hdot}
\end{equation}%
In particular, once $H$ reaches its upper limit, further investment in the
technology would be worthless and we should have $j=0$. The parameter $\psi $
determines how investment in research enhances the accumulation of knowledge from experience.

As with fossil fuel, for the renewable backstop technology to be productive
on net, we require $p<1$. In effect, the renewable technology combines some
output (effectively, capital) with an exogenous energy source (for example,
sunlight, wind, waves or stored water) to produce more useful output than
has been used as an input.

\subsection{The Optimization Problem}

Goods are consumed, invested in $k$ or $H$, or used for producing fossil
fuel or backstop energy input. This leads to a resource constraint (in per capita
terms): 
\begin{equation}
c+i+j+n+g(S,N)R+pB=y  \label{eq:Budget}
\end{equation}

The objective function is maximized subject to the differential constraints %
\eqref{eq:Sdot}, \eqref{eq:Ndot}, \eqref{eq:kdot} and \eqref{eq:Hdot} with
initial conditions $S(0)=N(0)=0$, $k(0)=k_{0}>0$ and $H(0)=0$, the budget
constraint \eqref{eq:Budget}, the definitions of output \eqref{eq:ProdFn},
energy input \eqref{eq:NrgInput} and the evolution of the cost of the
backstop energy supply \eqref{eq:RenewCost}. The control variables are $%
c,i,j,R,n$ and $B$, while the state variables are $k,H,S$ and $N$. Denote
the corresponding co-state variables by $q,\eta ,\sigma $ and $\nu $. Let $%
\lambda $ be the Lagrange multiplier on the budget constraint. We also need
to allow for the possibility that either type of energy source is not used
and investment in cost reduction for the energy technology is zero. To that
end, let $\mu $ the multiplier on the constraint $j\geq 0$, $\omega $ the
multiplier on the constraint $n\geq 0$, $\xi $ the multiplier on the
constraint $R\geq 0$ and $\zeta $ the multiplier on the constraint $B\geq 0$%
. Finally, let $\chi $ be the multiplier on the constraint $H\leq \Gamma
_{2}{}^{-1/\alpha }-$$\Gamma _{1}$ on the accumulation of knowledge about
the renewable technology.

Define the current value Hamiltonian and thus Lagrangian by 
\begin{equation}
\begin{split}
& \mathcal{H}=\frac{c^{1-\gamma }}{1-\gamma }+\lambda \left[
Ak-c-i-j-n-g(S,N)R-(\Gamma _{1}+H)^{-\alpha }B\right] +\epsilon (R+B-Ak) \\
& +q(i-\delta k)+\eta B(1+\psi j)+\sigma QR+\nu n+\mu j+\omega n+\xi R+\zeta
B+\chi \lbrack \text{$\Gamma _{2}^{-1/\alpha }-$}\Gamma _{1}-H]
\end{split}
\label{eq:Hamiltonian}
\end{equation}

The first order conditions for a maximum with respect to the control
variables are: 
\begin{equation}
\frac{\partial \mathcal{H}}{\partial c}= c^{-\gamma} -\lambda = 0
\label{eq:FOCc}
\end{equation}
\begin{equation}
\frac{\partial \mathcal{H}}{\partial i}= -\lambda + q = 0  \label{eq:FOCi}
\end{equation}
\begin{equation}
\frac{\partial \mathcal{H}}{\partial j}=-\lambda + \eta\psi B + \mu = 0;
\mu j=0, \mu\ge 0, j \ge 0  \label{eq:FOCj}
\end{equation}
\begin{equation}
\frac{\partial \mathcal{H}}{\partial n}= -\lambda +\nu+\omega = 0, \omega
n=0, \omega\ge 0, n\ge 0  \label{eq:FOCn}
\end{equation}
\begin{equation}
\frac{\partial \mathcal{H}}{\partial R}= -\lambda g(S,N)+\epsilon+\sigma
Q+\xi = 0, \xi R=0, \xi\ge 0, R\ge 0  \label{eq:FOCR}
\end{equation}
\begin{equation}
\frac{\partial \mathcal{H}}{\partial B}= -\lambda
(\Gamma_1+H)^{-\alpha}+\epsilon+\eta(1+\psi j)+\zeta=0, \zeta B=0, \zeta\ge 0, B\ge 0
\label{eq:FOCB}
\end{equation}

The differential equations for the co-state variables are: 
\begin{equation}
\dot{q}=\beta q-\frac{\partial \mathcal{H}}{\partial k}=(\beta +\delta
)q-\lambda A+\epsilon A  \label{eq:qdot}
\end{equation}
\begin{equation}
\begin{split}
&\dot{\eta} =\beta \eta -\frac{\partial \mathcal{H}}{\partial H}=\beta \eta
-\lambda \alpha (\Gamma _{1}+H)^{-\alpha -1}B+\chi ; \\&
\chi \lbrack (\text{$\Gamma _{2}^{-1/\alpha }-$}\Gamma _{1}-H] =0,\chi \geq
0,H\leq \text{$\Gamma _{2}^{-1/\alpha }-$}\Gamma _{1}
\end{split}
\label{eq:etadot}
\end{equation}
\begin{equation}
\dot{\sigma}=\beta \sigma -\frac{\partial \mathcal{H}}{\partial S}=\beta
\sigma +\lambda \frac{\partial g}{\partial S}R  \label{eq:sigmadot}
\end{equation}%
\begin{equation}
\dot{\nu}=\beta \nu -\frac{\partial \mathcal{H}}{\partial N}=\beta \nu
+\lambda \frac{\partial g}{\partial N}R  \label{eq:nudot}
\end{equation}%
We also recover the budget constraint \eqref{eq:Budget} and the differential
equations for the state variables, \eqref{eq:kdot}, \eqref{eq:Hdot}, %
\eqref{eq:Sdot} and \eqref{eq:Ndot}.

\subsection{The Long Run Endogenous Growth Economy}

Since the costs of using fossil fuel must rise as resources are depleted,
ultimately energy is supplied using only the backstop renewable technology.
In the very long run, the cost of the renewable energy source will be
constant at $p=\Gamma _{2}$ and the stock of knowledge about renewable
energy production $H$ is no longer relevant. In this regime, the model
becomes a simple endogenous growth model with investment only in physical
capital. We retain the first order conditions \eqref{eq:FOCc}, %
\eqref{eq:FOCi} and \eqref{eq:FOCB}, the first co-state equation %
\eqref{eq:qdot}, the budget constraint \eqref{eq:Budget} and the
differential equation \eqref{eq:kdot} for the only remaining state variable $%
k$. However, \eqref{eq:FOCB} changes to simply $\epsilon =\lambda \Gamma
_{2} $. From \eqref{eq:FOCi} we will obtain $q=\lambda $ and hence $\dot{q}=%
\dot{\lambda}$, and the co-state equation \eqref{eq:qdot} becomes 
\begin{equation}
\dot{\lambda}=\left[ \beta +\delta -(1-\Gamma _{2})A\right] \lambda \equiv 
\bar{A}\lambda  \label{eq:term_lam_dot}
\end{equation}%
where $\bar{A}$ is a constant. If we are to have perpetual growth, we must
have $c\rightarrow \infty $ as $t\rightarrow \infty $, which from %
\eqref{eq:FOCc} will require $\lambda \rightarrow 0$ and hence $\bar{A}<0$,
that is\footnote{%
Note that \eqref{eq:Gam2Cond1} will require $A>(\beta +\delta )/(1-\bar{p}%
)>\beta +\delta $, which is the usual condition for perpetual growth in a
simple linear growth model.} 
\begin{equation}
\Gamma _{2}<1-\frac{\beta +\delta }{A}  \label{eq:Gam2Cond1}
\end{equation}%
Condition \eqref{eq:Gam2Cond1} has an intuitive interpretation. With $B=y$ and $p=\Gamma_2$, $A(1-\Gamma_2)$ equals output per unit of capital \textit{net} of the costs of supplying the backstop energy input. To obtain perpetual growth, this must exceed the cost of holding capital measured by the sum of the depreciation rate (the explicit cost) and the time discount rate (the implicit opportunity cost). Hereafter, we assume \eqref{eq:Gam2Cond1} to be valid. The solution to %
\eqref{eq:term_lam_dot} can be written 
\begin{equation}
\lambda =\bar{K}e^{\bar{A}t}  \label{eq:term_lam_sol}
\end{equation}%
for some constant $\bar{K}$ yet to be determined. Thus, in this final
regime, the budget constraint, the first order condition \eqref{eq:FOCc} for 
$c$ and \eqref{eq:term_lam_sol} imply 
\begin{equation}
\dot{k}=(\beta -\bar{A})k-\bar{K}^{-1/\gamma }e^{-\bar{A}t/\gamma }
\label{eq:TermBudget}
\end{equation}%
The integrating factor for the differential equation \eqref{eq:TermBudget}
is $e^{(\bar{A}-\beta )t}$, so the solution can be written 
\begin{equation}
k=C_{0}e^{(\beta -\bar{A})t}+\frac{\gamma \bar{K}^{-1/\gamma }e^{-\bar{A}%
t/\gamma }}{\beta \gamma -\bar{A}(\gamma -1)}  \label{eq:Term_k_gensol}
\end{equation}%
for another constant $C_{0}$. However, the transversality condition at
infinity requires 
\begin{equation}
\lim_{t\rightarrow \infty }e^{-\beta t}\lambda k=C_{0}+\lim_{t\rightarrow
\infty }\frac{\gamma \bar{K}^{-1/\gamma }e^{(-\bar{A}/\gamma +\bar{A}-\beta
)t}}{\beta \gamma -\bar{A}(\gamma -1)}=0  \label{eq:TVC}
\end{equation}%
Equation \eqref{eq:TVC} in turn requires 
\begin{equation}
C_{0}=0\,\text{ and }\,\bar{A}(\gamma -1)<\beta \gamma  \label{eq:TVCresult}
\end{equation}%
Note that since $\bar{A}<0$ the inequality in \eqref{eq:TVCresult} will be satisfied if $%
\gamma >1$, while if $0<\gamma <1$, it will require 
\begin{equation}
\Gamma _{2}>1-\frac{\beta /(1-\gamma )+\delta }{A}  \label{eq:Gam2Cond2}
\end{equation}%
Thus, for $\gamma <1$, the validity of \eqref{eq:Gam2Cond1} and %
\eqref{eq:Gam2Cond2} together require

\begin{equation}
1-\frac{\beta +\delta }{A}>\Gamma _{2}>1-\frac{\beta /(1-\gamma )+\delta }{A}
\label{eq:pbarConstr}
\end{equation}

In summary, we conclude that the value of $k$ in the final endogenous growth
economy will be given by 
\begin{equation}
k = \frac{\gamma \bar{K}^{-1/\gamma}e^{-\bar{A}t/\gamma}}{\beta\gamma-\bar{A}%
(\gamma-1)}  \label{eq:Term_k_sol}
\end{equation}
with $\lambda$ given by \eqref{eq:term_lam_sol} and $\bar{K}$ is a constant
yet to be determined.

For periods prior to the terminal endogenous growth regime just analyzed, note first that we cannot have $j>0$ and $B=0$.  This follows from \eqref{eq:FOCj}, since if $B=0$, $\mu=\lambda>0$ which implies $j=0$.  For empirically relevant parameter values, however, we can have a short interval of time where $B>0$ and $j=0$. Since $B>0$ in this regime, learning by doing implies that the cost of renewable energy will decline.

Since the energy services of fossil fuels and the backstop renewable technology are perfect substitutes, the renewable technology will not be competitive with fossil fuels until the shadow price of energy in the fossil fuel regime equals the shadow price in the renewable backstop regime. When only fossil fuel is used, we assume that the productivity of investing in fossil fuel technology is high enough to sustain investments right up until the time the economy transitions to renewable energy. Although investments moderate the increase in fossil fuel costs, eventually depletion ensures that the shadow price of energy derived from fossil fuels rises to equal the initially higher cost of energy from renewable sources. At that point, the economy switches to use only renewable energy and all investment in, and use of, fossil fuel technologies ceases. We therefore conclude that the economy will pass through the regimes illustrated in the time line in Figure \ref{fig:Regimes}.

\begin{figure}[ht]
\centering \includegraphics[width=6.5in]{FourRegime.pdf}
\caption{Regimes of energy use and investment}
\label{fig:Regimes}
\end{figure}

\subsection{The Initial Fossil Fuel Economy}

It is useful to consider next the regime where $R>0$. Then \eqref{eq:FOCR} implies $\xi =0$ and the shadow price of energy
will be 
\begin{equation}
\epsilon =\lambda g(S,N)-\sigma Q  \label{eq:NrgP_fossregime}
\end{equation}%
Since an increase in $S$ raises the costs of fossil fuel, the co-state
variable $\sigma $ will be negative\footnote{%
Recall that if we use $V$ to denote the maximized value of the objective subject to the
constraints, the co-state variable $\sigma $ will equal the partial
derivative of $V$ with respect to the corresponding state variable, $S$.}
while fossil fuels are used as an energy source. It then follows from %
\eqref{eq:NrgP_fossregime} that the shadow price of energy $\epsilon $ is
unambiguously positive.

We also assume parameter values are chosen so that investment in fossil fuel technology is productive, that is, $n>0$. Then \eqref{eq:FOCn} implies $\omega =0$ and hence $\nu =\lambda $. But then $\dot{\nu}=\dot{\lambda}$ and \eqref{eq:nudot} implies
\begin{equation}
\dot{\lambda}=\beta\lambda+\lambda\frac{\partial g}{\partial N}R
\label{eq:lamdotfoss}
\end{equation}
If we also have $i>0$, \eqref{eq:FOCi} will imply $\lambda=q$ and from \eqref{eq:qdot} and \eqref{eq:NrgP_fossregime}, we will also have $\dot{\lambda}=(\beta+\delta+g(S,N)A-A)\lambda-\sigma QA$. Using \eqref{eq:lamdotfoss} we then conclude
\begin{equation}
\left[ \delta +g(S,N)A-\frac{\partial g}{\partial N}R-A\right] \lambda
=\sigma QA  \label{eq:qdoteqnudot}
\end{equation}%
Note that since $\sigma <0$ and $\lambda =c^{-\gamma }>0$, a necessary
condition for \eqref{eq:qdoteqnudot} to hold is that 
\begin{equation}
\delta +g(S,N)A-\frac{\partial g}{\partial N}R<A  \label{eq:FossNC}
\end{equation}%
Since we have assumed, however, that $g(S,N)$ eventually increases above $1$
as $S$ grows, and $\partial g/\partial N<0$, constraint \eqref{eq:FossNC}
must eventually be violated and the economy will not use fossil fuels
forever.

Substituting $R=Ak$ into \eqref{eq:qdoteqnudot}, we obtain an equation relating $N$ and $k$. When there is active investment in two types of capital (here $k$ and $N$), the investment has to maintain a relationship between the two stocks. Differentiating the resulting expression with respect to time, substituting for $\dot{N},\dot{\lambda}/\lambda =\dot{\nu}/\nu ,\dot{S},\dot{\sigma}$ and $\dot{Q}=\pi Q$ (since the exogenous growth rate of $Q$ is $\pi $), and using \eqref{eq:qdoteqnudot}, we obtain a condition relating the two types of investments ($i$ and $n$) in the initial fossil fuel economy: 
\begin{equation}
\lambda \left[ \frac{\partial g}{\partial N}(n+\delta k+\frac{\sigma QAk}{%
\lambda }-i)-\frac{\partial ^{2}g}{\partial S\partial N}QAk^{2}-\frac{%
\partial ^{2}g}{\partial N^{2}}nk\right] =\sigma \pi Q  \label{eq:i_n}
\end{equation}

We obtain a second relationship from the budget constraint. Specifically, using the result that $j=0$ if $B=0$, the first order condition \eqref{eq:FOCc}
for $c$, the production function \eqref{eq:ProdFn}, the energy input demand
requirement \eqref{eq:NrgInput} the budget constraint \eqref{eq:Budget} implies: 
\begin{equation}
i=Ak[1-g(S,N)]-\lambda ^{-1/\gamma }-n  \label{eq:i_fossregime}
\end{equation}%
Substituting \eqref{eq:i_fossregime} into \eqref{eq:i_n}, we then obtain an
equation to be solved for energy technology investment $n$ in the fossil
fuel regime: 
\begin{equation}
\begin{split}
& n\lambda\left( \frac{\partial ^{2}g}{\partial N^{2}}k-2\frac{\partial g}{%
\partial N}\right) =\\&\lambda \left[ \frac{\partial g}{\partial N}[k(\delta
+g(S,N)A-A+\frac{\sigma QA}{\lambda })+\lambda ^{-1/\gamma }]-\frac{\partial
^{2}g}{\partial S\partial N}QAk^{2}\right] -\sigma \pi Q  
\end{split}
\label{eq:n_soln}
\end{equation}%
Since $\partial g/\partial N<0$ and $\partial ^{2}g/\partial N^{2}>0$, the
coefficient of $n$ on the left hand side of \eqref{eq:n_soln} is positive.
From the budget constraint \eqref{eq:i_fossregime}, $\delta
k+Ak(g-1)+\lambda ^{-1/\gamma }=\delta k-i-n\le\delta k-n$. Then if
\begin{equation}
-\frac{\partial^{2}g}{\partial S\partial N}QAk^{2}+\frac{\partial g}{\partial N}(\delta+\frac{\sigma QA}{\lambda }) k-\sigma \pi Q>0
\label{eq:nposcond}
\end{equation}
we can conclude that $n>0$ as hypothesized.\footnote{Since $\partial^{2}g/\partial S\partial N<0$ and $\sigma<0$, the quadratic in $k$ in \eqref{eq:nposcond} has a positive second derivative and positive intercept, so even if $\delta+\sigma QA/\lambda>0$, so the roots are both positive, we conclude that \eqref{eq:nposcond} must hold for large $k$. For small values of $k$, we are likely to have $\dot{k}=i-\delta k>0$, in which case the right hand side of \eqref{eq:n_soln} is guaranteed to be positive.} Using the solution for $n$ and the current values of the state and co-state variables, \eqref{eq:i_fossregime} can be solved for $i$.

In summary, we conclude that the initial period of fossil fuel use with both $i>0$ and $n>0$ produces five differential equations
for $k$, $S$, $N$, $\sigma $, and $\lambda $: 
\begin{equation}
\dot{k}=i-\delta k 
\label{eq:Reg1_keq}
\end{equation}
\begin{equation}
\dot{S}=QAk 
\label{eq:Reg1_Seq}
\end{equation}
\begin{equation}
\dot{N}=n 
\label{eq:Reg1_Neq}
\end{equation}
\begin{equation}
\dot{\sigma}=\beta \sigma +\lambda \frac{\partial g}{\partial S}Ak
\label{eq:Reg1_sigeq}
\end{equation}
\begin{equation}
\dot{\lambda}=\lambda(\beta+\delta+(g(S,N)-1)A)-\sigma QA
\label{eq:Reg1_lameq}
\end{equation}
together with the exogenous population growth $Q=Q_{0}e^{\pi t}$.

\subsection{The Intermediate Economy with Renewables and Technological
Progress}

We next consider the regimes where $B=Ak>0,j\ge 0$ and $H<\Gamma _{2}^{-1/\alpha}- \Gamma _{1}$. For $B>0$, \eqref{eq:FOCB} implies $\zeta =0$, while $H<\Gamma _{2}^{-1/\alpha}- \Gamma _{1}$ and \eqref{eq:etadot} imply $\chi =0$. Considering first the majority of this regime where $j>0$,  \eqref{eq:FOCj} implies $\mu =0$, and from \eqref{eq:FOCi} and \eqref{eq:FOCj},  $q=\lambda =\eta\psi Ak$. Thus, when $j>0$ the shadow price of energy becomes

\begin{equation}
\epsilon =\lambda (\Gamma _{1}+H)^{-\alpha }-\frac{\lambda(1+ \psi j)}{\psi Ak}
\label{eq:BstopNrgP}
\end{equation}%
Substituting \eqref{eq:BstopNrgP} into \eqref{eq:qdot} and noting that $%
q=\lambda $ implies $\dot{q}=\dot{\lambda}$, we obtain 
\begin{equation}
\frac{\dot{\lambda}}{\lambda }=\beta +\delta -A(1-(\Gamma _{1}+H)^{-\alpha
})-\frac{1}{\psi k}-\frac{j}{k}  \label{eq:lamdotreg2}
\end{equation}%
From \eqref{eq:etadot} with $\lambda =\eta\psi Ak$ and $B=Ak$, and
using $\dot{k}=i-\delta k$, we obtain 
\begin{equation}
\frac{\dot{\lambda}}{\lambda }=\beta-\delta -\alpha (\Gamma _{1}+H)^{-\alpha
-1}\psi (Ak)^2+\frac{i}{k}
\label{eq:etadotreg2}
\end{equation}%
Equating \eqref{eq:lamdotreg2} and \eqref{eq:etadotreg2}, we obtain an
expression for total investment, $i+j$, as a function of $k$
and $H$ 
\begin{equation}
i+j=2\delta k-\frac{1}{\psi}-Ak(1-(\Gamma _{1}+H)^{-\alpha})+\alpha \psi A^2k^{3}(\Gamma _{1}+H)^{-\alpha -1}  \label{eq:Invreg2}
\end{equation}%
The budget constraint and the first order condition \eqref{eq:FOCc} for $c$
then provide a second equation for $i+j$: 
\begin{equation}
i+j=Ak(1-(\Gamma _{1}+H)^{-\alpha })-\lambda ^{-1/\gamma }
\label{eq:budgetreg2}
\end{equation}%
Substituting \eqref{eq:budgetreg2} into \eqref{eq:Invreg2}, we obtain an
equation relating $H$ and $k$: 
\begin{equation}
\alpha \psi A^2k^{3}(\Gamma _{1}+H)^{-\alpha -1}+2k[\delta-A(1-(\Gamma _{1}+H)^{-\alpha })]+\lambda ^{-1/\gamma }-\frac{1}{\psi}=0
\label{eq:Hkreg2}
\end{equation}%
Once again, when there is active investment in two types of capital (here $k$ and $H$), the investment has to maintain a relationship between the two stocks.\footnote{It can be shown that \eqref{eq:Hkreg2} has a unique real solution for $k$ in terms of $H$ and the parameters.}

Differentiating \eqref{eq:Hkreg2} with respect to $t$, and substituting $%
\dot{k}=i-\delta k$, $\dot{H}=Ak(1+\psi j)$ and for $\dot{\lambda}/\lambda $
using \eqref{eq:lamdotreg2} we obtain a second relationship between $i$ and $%
j$ and the current values of $k,H$ and $\lambda $:

\begin{equation}
\begin{split}
& \left[ 2[\delta -A(1-(\Gamma _{1}+H)^{-\alpha })]+3\alpha\psi (Ak)^2(\Gamma_{1}+H)^{-\alpha -1}\right] i+ \\
& \left[ \frac{\lambda ^{-1/\gamma }}{\gamma k}-\alpha\psi (Ak)^2(\Gamma _{1}+H)^{-\alpha -1}[2+(1+\alpha)Ak^2(\Gamma _{1}+H)^{-1}]\right] j \\
& =\left[ 2[\delta -A(1-(\Gamma _{1}+H)^{-\alpha })]+3\alpha\psi (Ak)^2(\Gamma_{1}+H)^{-\alpha -1}\right] \delta k \\
&+\alpha (Ak)^2(\Gamma _{1}+H)^{-\alpha -1}[2+(1+\alpha)Ak^2(\Gamma _{1}+H)^{-1}]\\
& +\frac{\lambda ^{-1/\gamma }}{\gamma }[\beta +\delta -A(1-(\Gamma_{1}+H)^{-\alpha })-\frac{1}{\psi k}]
\end{split}
\label{eq:ijrelreg2}
\end{equation}%
The two equations \eqref{eq:budgetreg2} and \eqref{eq:ijrelreg2} can then be
solved for $i$ and $j$ given current values for $k,H$ and $\lambda $. Using
the solutions for the investment levels, we can then obtain the differential
equations governing the evolution of $k,H$ and $\lambda $ in region 2,
which, for convenience are summarized below: 
\begin{gather}
\dot{k}=i-\delta k  \label{eq:Reg2_keq} \\
\dot{H}=Ak(1+\psi j)  \label{eq:Reg2_Heq}\\
\dot{\lambda}=\lambda \left[ \beta +\delta -A(1-(\Gamma _{1}+H)^{-\alpha })-\frac{1+\psi j}{\psi k}\right]  \label{eq:Reg2_lameq}
\end{gather}

Now consider the beginning of the renewable energy regime where $j=0$. Here we will have from \eqref{eq:FOCj} that $\mu=\lambda-\eta\psi B\ge 0$ with $\lambda=\eta\psi B$ at the upper boundary where the constraint on $j$ is just binding. In the interior of this region where $n=j=0=R$, the budget constraint \eqref{eq:Budget} will imply $i=Ak(1-p)-c$ with $p=(\Gamma_1+H)^{-\alpha}$ and $c=\lambda^{-1/\gamma}$. Also, the shadow price of energy obtained from \eqref{eq:FOCB} will now be given by $\epsilon=\lambda p-\eta$ and we retain $q=\lambda$. However, we will no longer have $\lambda=\eta\psi Ak$ and $\eta$ and $\lambda$ will evolve separately. The differential equations governing the evolution of $k,H,\lambda$ and $\eta $ now become: 
\begin{gather}
\dot{k}=Ak[1-(\Gamma_1+H)^{-\alpha}]-\lambda^{-1/\gamma}-\delta k  \label{eq:Reg2a_keq} \\
\dot{H}=Ak  \label{eq:Reg2a_Heq} \\
\dot{\lambda}=\lambda \left[ \beta +\delta -A(1-(\Gamma _{1}+H)^{-\alpha })\right]-\eta A  \label{eq:Reg2a_lameq}\\
\dot{\eta}=\beta\eta-\lambda\alpha(\Gamma _{1}+H)^{-\alpha-1}Ak \label{eq:Reg2a_etaeq}
\end{gather}

\subsection{Boundary conditions}

In the numerical analysis, the economy begins with known values of the state variables $k(0),S(0)$ and $N(0)$ at $t=0$. However, the initial values of the co-state variables $\lambda(0)$ and $\sigma(0)$ are unknown. Similarly, the initial value of the co-state variable $\eta(T_0)$ at $T_0$ is unknown. These all have to be guessed and the model solved forward. The values of the co-state variables at the transition times are then compared with their target values and the guesses are modified until all the targets are attained to the desired numerical accuracy. In this section, we discuss what the target values ought to be.

First, note that at $T_0$, $H=j=0=\sigma$. Using \eqref{eq:BstopNrgP} and \eqref{eq:NrgP_fossregime}, the fact that and the requirement that the shadow price of energy has to be continuous across the region boundaries then implies 
\begin{equation}
\Gamma _{1}^{-\alpha }-\frac{\eta}{\lambda}=\frac{\epsilon}{\lambda}= g(S,N) 
\label{eq:T_0NrgP}
\end{equation}
For a given value of $\eta(T_0)$, \eqref{eq:T_0NrgP} would then determine a value of $T_0$ and corresponding values of $k(T_0),S(T_0),N(T_0),\sigma(T_0)$ and $\lambda(T_0)$  using the differential equations \eqref{eq:Reg1_keq}--\eqref{eq:Reg1_lameq}. The calculated value of $\sigma(T_0)$ would then need to be compared to its target value of 0.

The calculated values for $k(T_0)$ and $\lambda(T_0)$ together with $H=0$ and the guessed value of  $\eta(T_0)$ will then provide starting values for the differential equations \eqref{eq:Reg2a_keq}--\eqref{eq:Reg2a_etaeq} in the next regime.\footnote{It may also be worth noting that a number of control variables will not be continuous across the $t=T_0$ boundary. To begin with, $R$ will jump from equalling $Ak>0$ right up until $T_0$ to a value of zero at $T_0$ and beyond. Correspondingly, $B$ will jump from zero before $T_0$ to $Ak>0$ from $T_0$ on. In addition, $n$ will jump from being strictly positive as $t\to T_0$ to being zero at $T_0$. Conversely, $j$ will jump from zero for $t<T_0$ to being positive at $T_0$.} As already noted above, the upper boundary $T_1$ of this region will occur where $\lambda=\eta\psi Ak$.

Once $T_1$ has been reached, the differential equations change to \eqref{eq:Reg2_keq}--\eqref{eq:Reg2_lameq}. Again, the values of $k,H$ and $\lambda$ will be continuous across the $T_1$ boundary. We also will require that the initial calculated value for $j$ in the third regime equal its target value of 0.  The upper boundary of this third region, $T_2$, will occur where $p=(\Gamma_{1}+H)^{-\alpha }=\Gamma _{2}$, which will determine the value of $H$ at $T_{2}$, namely $H=\Gamma_{2}^{-1/\alpha }-\Gamma _{1}$. We will also know the values of $k$ and $\lambda $ at $T_{2}$ (up to the unknown constant $\bar{K}$) since they must be continuous across the boundary and therefore must equal \eqref{eq:Term_k_sol} and \eqref{eq:term_lam_sol} respectively. One of these equations, say \eqref{eq:term_lam_sol}, can be used to solve for $\bar{K}$ and  then \eqref{eq:Term_k_sol} will provide a third target value.

Note that in total we have three targets $\sigma(T_0)=0,j(T_1)=0$ and $k(T_2)$ equals the corresponding calculated value implied by $\lambda(T_2)$, that can be used to determine appropriate initial values for the three variables $\lambda(0),\sigma(0)$ and $\eta(T_0)$. In practice, we guess values for the latter and iterate until the targets are attained.

\subsection{Calibration}

In order to quantitatively evaluate different policy scenarios, we first
need to calibrate the theoretical model. This involves assigning numerical
values to certain parameters in a way that make the model consistent with
observations from the actual world economy. By definition, we start the
economy with $S=N=H=0$ and with $Q=Q_{0}$. For convenience, we take the
current population $Q_{0}=1$ and effectively measure future population as
multiples of the current level. We will assume that the population growth
rate is 1\%.\footnote{%
This is consistent with a simple extrapolation of recent world growth rates
reported by the Food And Agriculture Organization of the United Nations, 
\textsf{http://faostat.fao.org/site/550/default.aspx}}

In line with standard assumptions made to calibrate growth models, we assume
a time discount factor $\beta =0.05$ and a coefficient of relative risk
aversion $\gamma =3$. However, as we explain in more detail below, we will allow $\gamma$ to adjust to ensure we match the initial share of consumption in GDP.

To calibrate values for the initial production,
capital and energy quantities we used data from the \textit{Energy
Information Administration} (EIA),\footnote{%
International data is available at \textsf{%
http://www.eia.doe.gov/emeu/international/contents.html}} the \textit{Survey
of Energy Resources 2007} produced by the \textit{World Energy Council},%
\footnote{%
This is available at \textsf{%
http://www.worldenergy.org/publications/survey\_of\_energy\_resources%
\_2007/default.asp} The data are estimates as of the end of 2005.} and 
\textit{The GTAP 7 Data Base} produced by the \textit{Center for Global
Trade Analysis} in the Department of Agricultural Economics, Purdue
University.\footnote{%
Information on this can be found at \textsf{%
https://www.gtap.agecon.purdue.edu/databases/v7/default.asp} The GTAP 7 data
base pertains to data for 2004.} The last mentioned data source is useful
for our purposes because it provides a consistent set of international
accounts that also take account of energy flows.

One of the first issues we need to address is that national accounts include
government spending in GDP, which does not appear in the model. In the GTAP
data base, government spending falls almost entirely (97\%) on the services
sector. We therefore believe that it is not unreasonable to classify it as
part of consumption spending in our model. Doing so gives an investment
share of expenditure of 0.21308.\footnote{%
Note that in the GTAP data base, aggregate world exports equal aggregate
world imports so world GDP equals consumption plus investment plus
government expenditure.} Effectively defining units so that aggregate output
is 1, we therefore identify 0.21308 as the sum $i+n$ at $t=0$. We would
expect most of this to be investment in capital used to produce output
rather than fossil fuel exploration and development.

Converting the GTAP data base estimates of the total capital stock capital
stock to units of GDP, we obtain the initial condition $k=2.98203$. We also use the GTAP
depreciation rate on capital of 4\%. Also, if we choose units so that output
equals 1, the parameter $A$ would equal the ratio of output to capital, that
is, $A\approx 0.33534$.

From the budget constraint, the difference between total output and the sum
of the investments, namely 0.78692 would equal consumption plus the current
costs $gR$ of supplying fossil fuels. We separate these two components using
sectoral data from the GTAP data base. Specifically, we classified
\textquotedblleft energy expenditure\textquotedblright\ as combined spending
on the primary fuels coal, oil and natural gas and the energy commodity
transformation sectors of refinery products, electricity
generation and natural gas distribution. The current cost of fossil energy
was then set equal to the expenditure on these sectors that was classified
as consumption or government spending rather than investment. This produced
a value for $gR=0.0284$. Then, after we set $\alpha_2$ and $\alpha_3$ so the initial values of $S$
and $N$ are zero, the initial value for $gR$ would imply 
\begin{equation}
\frac{0.0284}{R}=\alpha _{0}+\frac{\alpha _{1}}{\bar{S}-S(0)-\alpha _{2}/(\alpha
_{3}+N(0))}
\label{eq:Init_g}
\end{equation}
where $S(0)$ and $N(0)$ are the calculated values of $S$ and $N$ at $t=0$, which need to equal the target values $N(0)=0=S(0)$ in a fully-calibrated solution. In addition, subtracting the initial value for $gR$ from 0.78692 we obtain the initial value of $c(0)=0.75852$, and thus an initial target value for $\lambda(0)=c(0)^{-\gamma}$. We will adjust the value of $\gamma$ to help us attain this targeted value of initial consumption.

We can obtain a value for total fossil fuel production, $R$, from the EIA web site. It gives world wide production of oil in 2004 of 175.948 quads (where one quad equals $10^{15}$ BTU), of natural gas 100.141 quads and of coal 116.6 quads. Summing these gives a total of 392.689 quads. We then choose energy units so that the initial value of $R=1$.

To obtain an estimate of total fossil fuel resources $\bar{S}$ in the same units, we begin
with the proved and estimated additional resources in place from the World
Energy Council. The millions of tonnes of coal, millions of barrels of oil,
extra heavy oil, natural bitumen and oil shale and trillions of cubic feet
of natural gas given in that publication were converted to quads using
conversion factors available at the EIA. The result is 115.2 quintillion
BTU, or almost 300 times the annual worldwide production of fossil fuels in
2004. These resources are nevertheless relatively small compared to
estimates of the volume of methane hydrates that may be available. Although
experiments have been conducted to test methods of exploiting methane
hydrates, a commercially viable process is yet to be demonstrated. Partly as
a result, resource estimates vary widely. According to the National Energy
Technology Laboratory (NETL),\footnote{\textsf{http://www.netl.doe.gov/technologies/oil-gas/FutureSupply/MethaneHydrates/about-hydrates/estimates.htm}} the United States Geological Survey (USGS) has
estimated potential resources of about 200,000 trillion cubic feet in the
United States alone. According to Timothy Collett of the USGS,\footnote{\textsf{http://www.netl.doe.gov/kmd/cds/disk10/collett.pdf}} current estimates of the
worldwide resource in place are about 700,000 trillion cubic feet of methane. Using the
latter figure, this would be equivalent to 719.6 quintillion BTU. Adding
this to the previous total of oil, natural gas and coal resources yields a
value for $\bar{S}=834.8$ quintillion BTU or around 2125.855 in terms of the energy
units defined so that $R=1$.

We still need to specify values for the $\alpha_i$ parameters in the $g$ function. Assuming we know the value of $g(S,N)$ and its first partial derivatives at $t=0$, we will have three equations to determine the four $\alpha$ parameters. However, under these assumptions, the $g$ function is not unambiguously determined by its value, and the value of its first derivatives, at $t=0$. The reason is that we can always choose the parameters $\alpha_0, \alpha_1, \alpha_2$ and $\alpha_3$ to ensure that the initial state of knowledge equals its target value $N(0)=0$, and the initial level of cumulative exploitation also equals its target value $S(0)=0$, while leaving the value of $g$, its first partial derivatives, and $\partial^2g/\partial S^2$ and $\partial^2g/\partial S\partial N$ unchanged.\footnote{While $\partial^2g/\partial N^2$ will generally change under this transformation, we do not have sufficient information to calibrate a value for the second partial derivatives. We have chosen a transformation that keeps the second partial derivatives involving $S$ unchanged.} To see this, suppose that at some initial values of $\alpha_0, \alpha_1, \alpha_2$ and $\alpha_3$, $N(0)=N_0\ne 0$ and $S(0)=S_0\ne 0$. Now choose $\tilde{\alpha_0}, \tilde{\alpha_1}, \tilde{\alpha}_2$ and $\tilde{\alpha}_3$ so that
\begin{equation}
\tilde{\alpha_0}+\frac{\tilde{\alpha_1}\tilde{\alpha_3}}{\bar{S}\tilde{\alpha_3}-\tilde{\alpha_2}}=\alpha_0+\frac{\alpha_1(\alpha_3+N_0)}{(\bar{S}-S_0)(\alpha_3+N_0)-\alpha_2}
\label{eq:NormAlpha0}
\end{equation}

\begin{equation}
\frac{\tilde{\alpha_1}\tilde{\alpha}_3^2}{(\bar{S}\tilde{\alpha}_3-\tilde{\alpha}_2)^2}=\frac{\alpha_1(\alpha_3+N_0)^2}{[(\bar{S}-S_0)(\alpha_3+N_0)-\alpha_2]^2}
\label{eq:NormAlpha1}
\end{equation}

\begin{equation}
\frac{\tilde{\alpha}_2}{\tilde{\alpha}_3^2}=\frac{\alpha_2}{(\alpha_3+N_0)^2}
\label{eq:NormAlpha2}
\end{equation}

\begin{equation}
\frac{\tilde{\alpha}_3}{\bar{S}\tilde{\alpha}_3-\tilde{\alpha}_2}=\frac{\alpha_3+N_0}{(\bar{S}-S_0)(\alpha_3+N_0)-\alpha_2}
\label{eq:NormAlpha3}
\end{equation}

Then we can conclude that if we have a solution to the model where $g$ and its partial derivatives have been calibrated to particular values but the initial values of $N(0)=N_0\ne 0$ and $S=S_0\ne 0$, we can re-define $\alpha_0, \alpha_1, \alpha_2$ and $\alpha_3$ as in \eqref{eq:NormAlpha0}, \eqref{eq:NormAlpha1}, \eqref{eq:NormAlpha2} and \eqref{eq:NormAlpha3} and the new solution path will have $N(0)=0=S(0)$. One can show that the equations \eqref{eq:NormAlpha0}, \eqref{eq:NormAlpha1}, \eqref{eq:NormAlpha2} and \eqref{eq:NormAlpha3} have the solutions $\tilde{\alpha}_0=\alpha_0$, $\tilde{\alpha}_1=\alpha_1$ and
\begin{equation}
\tilde{\alpha}_2=\alpha_2\bigl [1+\frac{S_0(\alpha_3+N_0)}{\alpha_2}\bigr ]^2
\label{eq:NewAlpha2}
\end{equation}
and
\begin{equation}
\tilde{\alpha}_3=\alpha_3+N_0+\frac{S_0(\alpha_3+N_0)^2}{\alpha_2}
\label{eq:NewAlpha3}
\end{equation}

Thus, if we solve the differential equations backwards in time we will obtain some values $N_0$ for $N$ and $S_0$ for $S$ at $t=0$. After adjusting the values of $\alpha_2$ and $\alpha_3$ according to  \eqref{eq:NewAlpha2} and \eqref{eq:NewAlpha3}, we should match the initial conditions $N(0)=0=S(0)$. With this transformation of the $g$ function we therefore can accommodate the initial conditions on $N$ and $S$.

To begin the iterations, we need an initial set of values for the $\alpha_i$ coefficients. Noting that we can interpret $\bar{S}-\alpha_2/\alpha_3$ as the initial level of fossil fuel extraction $S$ at which marginal costs of extraction $g(S,0)$ would become unbounded, we associate  $\bar{S}-\alpha_2/\alpha_3$ with current proved and connected reserves of fossil fuel.\footnote{Note that current official reserves are not the relevant measure since many of these are not connected and thus are unavailable for production without further investment, denoted $n$ in the model.} A recent report from Cambridge Energy Research Associates (CERA, 2009),\footnote{\textquotedblleft The Future of Global Oil Supply: Understanding the Building Blocks,\textquotedblright\ Special Report by Peter Jackson, Senior Director, IHS Cambridge Energy Research Associates, Cambridge, MA.} for example, gives weighted average decline rates for oil production from existing fields of around 4.5\% per year. They also note that this figure is dominated by a small number of \textquotedblleft giant\textquotedblright\ fields and that, \textquotedblleft the average decline rate for fields that were actually in the decline phase was 7.5\%, but this number falls to 6.1\% when the numbers are production weighted.\textquotedblright\ Hence, we shall use 6\% as a decline rate for oil fields. If we use United States production and reserve figures as a guide, we find that natural gas decline rates are closer to 8\% per year but coal mine decline rates are closer to 6\% per year. In accordance with these figures, we assume the ratio of fossil fuel production to proved and connected reserves equals the share weighted average of these figures, namely $(175.948\ast 0.06+100.141\ast 0.08+116.6\ast 0.06)/392.689=0.0651$. Thus, in terms of the energy units defined so that $R=1$, the initial target value of $\bar{S} -\alpha_2/\alpha_3$ would equal 1/0.0651=15.361. Using the previously calculated value for $\bar{S}$, this leads to $\alpha_2/\alpha_3=2110.5$.

We also need to calibrate the partial derivatives of $g$ at $t=0$. Using GTAP data on capital shares by sector, we estimate that around 3\% of annual investment occurs in the oil, natural gas and coal sectors. We noted above that in the GTAP data, total investment $i+n=0.213$, implying that $n\approx 0.006$. We assume that this level of investment currently is sufficient to replace mined resources and allow for growth. Specifically, with $\alpha_2/\alpha_3=2110.5$, we assume that $\alpha_2/(\alpha_3+0.006)=2109$, which implies $\alpha_3\approx 8.44$. The previously calculated value for $\alpha_2/\alpha_3$ then implies a value for $\alpha_2 \approx 17816.7$. Given values for $\alpha_2$ and $\alpha_3$, the ratio $g_N/g_S$ then also is determined, but the individual values of $g_S$ and $g_N$ can still vary. As they do, $\alpha_0$ and $\alpha_1$ also will vary. Figure \ref{fig:MiningMCcurves} illustrates the curves for values of $g_S$ ranging from 0.00002 (the closest to a right angled shape) to 0.0004 (the furthest from a right angled shape). 

\begin{figure}[ht]
\centering \includegraphics[width=4in]{MiningMCcurves.pdf}
\caption{$g(S,N)$ for $N=0.01$ and different values of $g_S$ and $g_N$}
\label{fig:MiningMCcurves}
\end{figure}

We experimented with these different curves and will present different results as a sensitivity analysis.

We also require $k(0)$ to match its calibrated value $k_0=2.98203$ from the data. Finally, we also had an initial target value for $\lambda(0)=0.75852^{-\gamma}$. Recall that the solutions in regime 2 depend on two constants $T_{2}$ and $\bar{K}$, but one of these can be solved in terms of the other. Hence, we have one free parameter $T_2$ but we need to match two initial values for $k$ and $\lambda(0)$ at $t=0$. This is not a problem theoretically, since the value of the co-state variable $\lambda(0)$ is free. The requirement that $\lambda$ take a particular value at $t=0$ arises only because we are attempting to match a calibrated consumption share of GDP. In the numerical analysis, we will also allow the coefficient of relative risk aversion $\gamma$ to adjust to allow us to attain the targeted consumption level.

Turning next to the learning curve \eqref{eq:RenewCost}, the literature
provides a range of estimates for $\alpha $. An online calculator provided
by NASA\footnote{Available at \textsf{http://cost.jsc.nasa.gov/learn.html}} gives a range of
learning percentages between 5 and 20\% depending on the industry. A
learning percentage of $x$, which corresponds to a value of $\alpha
=-ln(1-x)/ln(2)$, has the interpretation that a doubling of the experience
measure will lead to a cost reduction of $x$\%. Thus, $x=0.2$ is equivalent
to $\alpha =0.322$ while $x=.05$ corresponds to $\alpha =0.074$. In a study
of wind turbines, Coulomb and Neuhoff (2006)\footnote{Louis Coulomb and Karsten Neuhoff, \textquotedblleft Learning Curves and
Changing Product Attributes: the Case of Wind Turbines\textquotedblright ,
University of Cambridge: Electricity Policy Research Group, Working Paper
EPRG0601.} found values of $\alpha $ of 0.158 and 0.197. In a 1998 paper, Gr\"{u}bler and Messner\footnote{Arnulf Gr\"{u}bler and Sabine Messner, \textquotedblleft Technological
change and the timing of mitigation measures\textquotedblright , \textit{Energy Economics} 20, 1998, 495--512} found a value for $\alpha =.36$ using
data on solar panels. In a 2008 paper in \textit{The Energy Journal}, van
Bentham et. al.\footnote{\textquotedblleft Learning-by-doing and the optimal solar policy in
California,\textquotedblright\ Arthur van Benthem, Kenneth Gillingham and
James Sweeney, 29(3) 2008, 131-152} report several studies finding a
learning percentage of around 20\% ($\alpha =0.322$) for solar panels. For
our base case, we will take $\alpha =0.37$.

The other parameter affecting the incentive to invest in renewable energy
sources is the initial value $\Gamma _{1}^{-\alpha }$ of the cost of using
renewable energy as the primary energy source. Using a document available
from the Energy Information Administration (EIA)\footnote{\textit{%
Assumptions to the Annual Energy Outlook, 2009}, \textquotedblleft
Electricity Market Module,\textquotedblright\ Table 8.2, available at 
\textsf{http://www.eia.doe.gov/oiaf/aeo/assumption/pdf/electricity.pdf%
\#page=3}} the cost of new onshore wind capacity is about double the cost of
combined cycle gas turbines (CCGT), while offshore wind is around four times
as expensive, solar thermal more than five times as expensive and solar
photovoltaic more than six times as expensive. However, these costs do not
take account of the lower average capacity factor of intermittent sources
such as wind or solar. The same document gives a fixed O\&M cost of onshore
wind that is around two and a half times the corresponding fixed O\&M for
CCGT, although the latter also has fuel costs. The corresponding ratio is
around 7 for offshore wind, while fixed O\&M for solar photovoltaic are
similar to the fixed O\&M for CCGT. As a rough approximation, we will assume 
$\Gamma _{1}^{-\alpha }=0.12$, which is a little over 4 times the initial
value of $g$. In accordance with the EIA assumptions, we also assume that,
in the long run, the renewable technologies can experience a five-fold
reduction in costs, so $\Gamma _{2}=0.024$. This would result in an energy
cost that is below the current cost of fossil fuel technologies.

Finally, we need to specify a value for $\psi $, the relative effectiveness
of direct investment in research versus learning by doing in accumulating
knowledge about new energy technologies. Klaassen et. al. (2005)\footnote{%
Klaassen, Ger, Asami Miketa, Katarina Larsen and Thomas Sundqvist,
\textquotedblleft The impact of R\&D on innovation for wind energy in
Denmark, Germany and the United Kingdom,\textquotedblright\ \textit{%
Ecological Economics}, 54 (2005) 227--240} estimated a model that allowed
for both learning-by-doing and direct R\&D. Although they assume the capital
cost is multiplicative in total R\&D and cumulative capacity, while we assume the 
\textit{change} in knowledge is multiplicative in new R\&D and cumulative
capacity, we can take their parameter estimates as a guide. They find direct
R\&D is roughly twice as productive for reducing costs as is
learning-by-doing.\footnote{%
Of course, the learning-by-doing has the advantage that it directly
contributes to output at the same time it is adding to knowledge.}
Consequently, we assume that $\psi =0.5$.

The results from the calibrated version of our model economy with $g_S=0.00005$ are summarized
below. Absent any government intervention in the economy, the transition to
a renewable energy regime occurs after $T_{1}=51$ years. The next benchmark,
when the renewable technology itself reaches its ultimate frontier, occurs
after $T_{2}=131$ years.

\begin{figure}[ht]
\centering \includegraphics[width=6.5in]{R1KP.pdf}
\caption{Fossil fuel regime without taxes or subsidies}
\label{fig:Regime1NoTaxes}
\end{figure}

Figure~\ref{fig:Regime1NoTaxes} shows the behavior of the main
variables in the economy during the initial regime. Fossil fuel use leads to
growth in consumption, as well as in the economy's capital stock. However,
increasing investment in the development of mining technologies is necessary
to meet demand as the economy grows. Towards the end of regime 1, the costs
associated with increase fossil fuel use are large. As a result, overall
investment in the economy declines, leading to a decrease to the rate of
growth of the economy's capital stock. This paves the way towards the
transition to the renewable energy regime.

\begin{figure}[ht]
\centering \includegraphics[width=6in]{R2KP.pdf}
\caption{Renewable regime without taxes or subsidies}
\label{fig:Regime2NoTaxes}
\end{figure}

Figure~\ref{fig:Regime2NoTaxes} shows the behavior of the main
variables in the first renewable energy regime. Here, economic
growth is fueled through the use of renewable energy. Direct investment in
renewable energy increases over time. Together with learning-by-doing, this
leads to the accumulation of technical knowledge that is necessary for a more
efficient use of this technology. Consumption and the economy's capital
stock continue to grow. Ultimately, a technological limit is reached, beyond
which there is no further decline in the cost of renewable energy.

\section{Policy Scenarios}

In this section we consider two alternative policies that could be used
to accelerate the adoption of renewable energy in the economy. The first
policy involves taxing investment in mining technology. This policy could
 keep the costs of using fossil fuel high, leading to an
acceleration of the adoption of the competing, renewable energy technology.
The second policy is a direct subsidy to R\&D expenditure in the
renewable energy sector. We examine several different subsidy rates in order to measure the sensitivity of the decline in the backstop energy price to the size of the subsidy.

\subsection{Scenario 1: Tax on Fossil Fuel Energy}

One way of indirectly subsidizing renewable energy might involve imposing a
tax on fossil fuels. Here, we consider different scenarios regarding the
size of such a tax and explore the implications for renewable technology
adoption and growth.

Introducing taxes on $n$ during Regime 1, the budget
constraint in that regime becomes:
\begin{equation}
c+i+j+n(1+\tau _{n})+g(S,N)R+pB=y+T
\end{equation}
with a corresponding budget constraint for the government given by:
\begin{equation}
\tau _{n}n=T
\end{equation}
Note that the revenue raised by the tax is returned in lump sum form. That is, when choosing investment in $n$, a private sector decision-maker takes account of the fact that higher $n$ implies a higher tax liability, but $T$ is taken as independent of any one individual's investment decision $n$.

The budget constraint in Regimes 2 and 3 is the same as before, so the
analysis of those regimes remains unchanged. For Regime 1, define the current value
Hamiltonian and thus Lagrangian by 
\begin{equation}
\begin{split}
\mathcal{H}=&\frac{c^{1-\gamma }}{1-\gamma }+\lambda \left[
Ak+T-c-i-j-n(1+\tau _{n})-g(S,N)R-(\Gamma _{1}+H)^{-\alpha }B\right] 
\\ & +\epsilon (R+B-Ak) 
+q(i-\delta k)+\eta j(1+\psi B)+\sigma QR+
\\& \nu n+\mu j+\omega n+\xi R+\zeta
B+\chi \lbrack \text{$\Gamma _{2}{}^{-1/\alpha }-$}\Gamma _{1}-H]
\end{split}
\end{equation}

The first order conditions for a maximum with respect to the control
variables are the same as previously except for $n$, which changes to: 
\begin{equation}
\frac{\partial \mathcal{H}}{\partial n}=-\lambda (1+\tau _{n})+\nu +\omega
=0,\omega n=0,\omega \geq 0,n\geq 0
\end{equation}%

The differential equations for the co-state variables remain as before. The
shadow price of energy will again be 
\begin{equation}
\epsilon =\lambda g(S,N)-\sigma Q
\end{equation}

As before, we also assume parameter values and taxes are chosen so that investment in
fossil fuel technology is productive, that is, $n>0$. Then $\omega =0$ and
hence $\nu =\lambda (1+\tau _{n})$, and, since $q=\lambda $, we have $\nu
=q(1+\tau _{n})$. But then using $\dot{q}=\dot{\lambda}$, and after substituting $R=Ak$, we now obtain 
\begin{equation}
\left[ \frac{\delta }{A}+g(S,N)-\frac{1}{(1+\tau _{n})}\frac{\partial g}{%
\partial N}k-1\right] \lambda =\sigma Q
\label{eq:qdoteqlamdotTaxes}
\end{equation}

Differentiating \eqref{eq:qdoteqlamdotTaxes} with respect to time, substituting for $\dot{N},\dot{\lambda}/\lambda =\dot{%
\nu}/\nu ,\dot{S},\dot{\sigma}$ and $\dot{Q}=\pi Q$, we obtain a condition
relating the two types of investments ($i$ and $n$) in the initial fossil
fuel economy: 
\begin{equation}
\lambda \left[ \frac{\partial g}{\partial N}\biggl (n(1+\tau_n)+\delta k+\frac{\sigma QAk}{%
\lambda }-i\biggr )-\frac{\partial ^{2}g}{\partial S\partial N}QAk^{2}-\frac{%
\partial ^{2}g}{\partial N^{2}}nk\right] =\sigma\pi Q(1+\tau_n)
\end{equation}

A second relationship between $i$ and $n$ is given by the budget constraint, which in this regime is:%
\begin{equation}
c+i+n(1+\tau _{n})+g(S,N)R=y+T
\label{eq:BudgetTaxes}
\end{equation}
In equilibrium, however, the government budget constraint will imply that per capita lump sum transfers equal per capita tax revenue. Also, $y=Ak=R$ and $c=\lambda^{-1/\gamma}$, so \eqref{eq:BudgetTaxes} can be written as:
\begin{equation}
i+n=Ak(1-g(S,N))-\lambda^{-1/\gamma}
\end{equation}


We consider different scenarios regarding the size of the tax. We summarize
our findings in Table~\ref{table:Tax}. The rows give the date of the transition to the
renewable energy regime ($T_{1}$), the cumulative investment in fossil fuel technology
at that time ($N$), the cumulative exploitation of fossil fuels before they are abandoned ($S$), and the date of transition to the final renewable energy
regime ($T_{2}$). The first column gives the outcome in the absence of government intervention. The next
three columns give the equilibrium values of the same variables when there
is a $2\%$, a $5\%$, and a $20\%$ tax on investment associated with fossil
fuel extraction.

\begin{table}[htdp]
\caption{Values of key variables with fossil fuel taxes}
\begin{center}\begin{tabular}{c|cccc}
  & $\tau _{n}=0 $& $\tau _{n}=0.02$ & $\tau _{n}=0.05$ & $\tau _{n}=0.2$\\
\hline
$T_{1}$ & 51.2249 & 46.3859 & 45.19 & 39.9463 \\
$N(T_{1})$ & 64.6412 & 58.0567 & 57.5293 & 55.1507 \\
$S(T_1)$ & 382.9009  & 350.9142 & 348.3918 & 334.1527\\
$T_{2}$ & 131.4168 & 126.5756 & 125.347 & 120.1413
\end{tabular}
\end{center}
\label{table:Tax}
\end{table}

These findings have a number of implications for policy. First, taxing
fossil fuels accelerates the rate of adoption of the renewable energy
technology. However, it is worth noting that the elasticity of the adoption
rate appears to be small. A tax as high as $20\%$ reduces $T_{1}$ by about $%
11$ years, while a more realistic tax of $2\%$ reduces $T_{1}$ by only $5$
years. The tax has the additional effect of decreasing investment in fuel
extraction. That is, the tax causes fossil fuel reserves to be used
less intensively in the fossil fuel economy in addition to accelerating the transition to renewable energy. Not surprisingly, therefore, the third row shows that the total extraction of fossil fuels also declines as a result of the tax. This outcome comes at some cost. The distortion created by
the tax creates a wedge between the equilibrium and the socially optimal
level of investment. Hence, it can be shown that social welfare in the
economy declines as a result of the tax.\footnote{%
Absent any government intervention, the \textit{First Welfare Theorem} holds
in our model economy. If, as a result of externalities or other distortions
the First welfare Theorem was to fail, then government policy could become
beneficial.}

\subsection{Scenario 2: Subsidy for Renewable Energy}

We are again interested in how effective the subsidy is in both bringing forward the time of the transition
to the renewable energy regime and also in reducing the total consumption of fossil fuels before that time is reached. Introducing a subsidy on $j$ during Regime 2, the budget constraint in that
regime becomes becomes:
\begin{equation}
c+i+j(1-\tau _{j})+n+g(S,N)R+pB=y-T
\end{equation}
with a corresponding budget constraint for the government given by:
\begin{equation}
\tau _{j}j=T
\end{equation}
Once again, the tax required to pay the subsidy is lump sum in the sense that individual decision-makers do not believe that their own choices of $j$ will affect the per capita tax bill.

The budget constraint in Regimes 1 and 3 is the same as before, so the
analysis of those regimes remains unchanged. For Regime 2, define the current value
Hamiltonian and thus Lagrangian by
\begin{equation}
\begin{split}
\mathcal{H}& =\frac{c^{1-\gamma }}{1-\gamma }+\lambda \left\{
Ak-T-c-i-j(1-\tau _{j})-n-g(S,N)R-(\Gamma _{1}+H)^{-\alpha }B\right\} \\
& +\epsilon (R+B-Ak)+q(i-\delta k)+\eta j(1+\psi B)+\sigma QR \\
& +\nu n+\mu j+\omega n+\xi R+\zeta B+\chi \lbrack \text{$\Gamma
_{2}{}^{-1/\alpha }-$}\Gamma _{1}-H]
\end{split}%
\end{equation}

The first order conditions for a maximum with respect to the control
variables once again are the same as before except for $j$ where the condition changes to: 
\begin{equation}
\frac{\partial \mathcal{H}}{\partial j}=-\lambda (1-\tau _{j})+\eta (1+\psi
B)+\mu =0;\mu j=0,\mu \geq 0,j\geq 0
\end{equation}%

The differential equations for the co-state variables remain as before.


Following the previous analysis, in regime 2 we will again have have $B=Ak>0$ and $j>0$. Now, however, $q=\lambda =\eta (1+\psi Ak)/(1-\tau _{j})$, so the shadow price of energy becomes 
\begin{equation}
\epsilon =\lambda(\Gamma_1+H)^{-\alpha }-\frac{\lambda \psi j(1-\tau _{j})}{%
1+\psi Ak}
\end{equation}%
Noting that $q=\lambda $ implies $\dot{q}=\dot{\lambda}$, we obtain 
\begin{equation}
\frac{\dot{\lambda}}{\lambda }=\beta +\delta -A(1-(\Gamma_1+H)^{-\alpha })-%
\frac{\psi Aj(1-\tau _{j})}{1+\psi Ak}
\label{eq:lamdotSubs1}
\end{equation}
Using $\lambda =\eta (1+\psi Ak)/(1-\tau _{j})$, $B=Ak$, and $\dot{k}=i-\delta k$, we obtain 
\begin{equation}
\frac{\dot{\lambda}}{\lambda }=\beta -\frac{\alpha (\Gamma_1+H)^{-\alpha
-1}Ak(1+\psi Ak)}{(1-\tau _{j})}-\frac{\psi A\delta k}{1+\psi Ak}+\frac{\psi
Ai}{1+\psi Ak}
\label{eq:lamdotSubs2}
\end{equation}
Equating \eqref{eq:lamdotSubs1} and \eqref{eq:lamdotSubs2}, we obtain an expression giving total
investment in regime 2, as a function of $k$ and $H$ 
\begin{equation}
\begin{split}
\psi A\left[ i+j(1-\tau _{j})\right] =& \delta (1+2\psi Ak)-A(1+\psi
Ak)(1-(\Gamma_1+H)^{-\alpha })
\\&+\frac{\alpha(\Gamma_1+H)^{-\alpha
-1}Ak(1+\psi Ak)^{2}}{(1-\tau _{j})}
\end{split}
\label{eq:ijeqnSubs}
\end{equation}
The budget constraint and the first order condition for $c$ then provide a
second equation: 
\begin{equation}
i+j=Ak(1-(\Gamma_1+H)^{-\alpha })-\lambda ^{-1/\gamma }
\label{eq:BudgetSubs}
\end{equation}%
where we have once again used the government budget constraint to eliminate the subsidy variable in equilibrium.

For $\tau_j\ne 0$, \eqref{eq:BudgetSubs} and \eqref{eq:ijeqnSubs} can now be solved for $i$ and $j$ as functions of $H, k$ and $\lambda$: 
\begin{equation}
(1+2\psi Ak)[\delta -A(1-(\Gamma_1+H)^{-\alpha })]+\frac{\alpha
(\Gamma_1+H)^{-\alpha -1}Ak(1+\psi Ak)^{2}}{(1-\tau _{j})}+\psi A\lambda
^{-1/\gamma}=-\psi A\tau_jj
\label{eq:jSubs}
\end{equation}
with $i$ then given from \eqref{eq:BudgetSubs}. Observe that the higher the subsidy rate $\tau_j$ the more negative has to be the left of \eqref{eq:jSubs}. In turn, since only the first term on the left of \eqref{eq:jSubs} can be negative, this will require a larger value of $H$ for given values of $k$ and $\lambda$. Not surprisingly, we conclude that a subsidy must increase investment in renewable technology knowledge $H$. With $H$ higher, the transition times must also come earlier in time under the subsidy policy.

As with the tax policy, we consider different scenarios regarding the size of the
subsidy. The first column in the Table~\ref{table:Subsidy} remains unchanged as it
gives the date of the transition to the renewable energy regime ($T_{1}$),
the cumulative investment in fossil fuel extraction at that time ($N$), the cumulative exploitation of fossil fuels at that time ($S$), and the date of
transition to the final renewable energy regime ($T_{2}$) in the absence of
any government intervention. The next three columns give the equilibrium
values of the same variables when there is a $2\%$, a $5\%$, and a $20\%$
subsidy on investment associated with renewable energy.

\begin{table}[htdp]
\caption{Values of key variables with renewable investment subsidies}
\begin{center}\begin{tabular}{c|cccc}
  & $\tau _{n}=0 $& $\tau _{n}=0.02$ & $\tau _{n}=0.05$ & $\tau _{n}=0.2$\\
\hline
$T_{1}$ & 51.2249 & 32.4124 & 24.0542 & 15.9956 \\
$N(T_{1})$ & 64.6412 & 87.1836 & 92.4245 & 110.4229 \\
$S(T_1)$ & 382.9009  & 478.2624 & 498.5666 & 566.7097 \\
$T_{2}$ & 131.4168 & 102.4820 & 90.0362 & 75.5973
\end{tabular}
\end{center}
\label{table:Subsidy}
\end{table}

These results contain some useful information for policy. First, a subsidy
on investment in renewable energy accelerates the rate of adoption of the
renewable energy technology. Indeed, although it is hard to compare the two
directly, a renewable energy subsidy appears to be more effective than a tax
on fossil fuels, with a $2\%$ subsidy accelerating $T_{1}$ by $19$ years.
Another important difference with the previous tax scenario is that the
fossil fuel reserves are used more intensively as a result of the subsidy.
The intuition of this result is as follows. Since the adoption of renewable
fuel is accelerated as a result of the subsidy, the opportunity cost of
using fossil fuel in the short run declines. Thus, while the subsidy on
renewables leads to a faster transition away from fossil fuels, it also
implies a more intensive use of fossil fuel than what is socially optimal in
the short run. While we do not model carbon dioxide or other emissions associated with the combustion of fossil fuels explicitly in our
analysis, it is worth mentioning that this could imply an increase in
such emissions in the short run.

\section{Conclusion}

With over two trillion dollars in annual sales, the energy industry is the
largest on the planet. Thus, economic policies that affect the energy sector
have global consequences. Yet, seldom are such policies studied and
evaluated using the standard tools of macroeconomics. In this paper we built
a model in which there is a potential for technological progress in
renewable energy to play the role of an engine of macroeconomic growth. We
computed the equilibrium optimal path of investment in both the fossil fuel and the
renewable energy sectors and calibrated the model to fit current global conditions using data from a variety of sources.
Finally, we evaluated different policy scenarios regarding imposing taxes on
the use of fossil fuel and offering government subsidies on the development
of renewable energy.

We found that, absent any government intervention, the economy goes through
three distinct regimes related to energy production. Initially, production
uses fossil fuel only, and investment takes place in order to improve the
efficiency of supplying fossil fuel. In the medium to long run, as the price
of fossil fuel inevitably increases, investment and capital accumulation
slow down. The economy then makes a transition to the first of two renewable
energy regimes. Subsequently, learning-by-doing reduces the cost of
producing capital using the backstop technology. Finally, in the very long
run, a transition to the second renewable energy scheme occurs. Here, a
limit is reached after which renewable energy is produced at the lowest
possible cost.

We then examined how these transitions are affected by imposing taxes on
fossil fuel and subsidies on R\&D in renewable energy. We found that taxing
fossil fuels accelerates the rate of adoption of the renewable energy
technology. However, a main finding of our analysis is that the elasticity
of the adoption rate appears to be small. In our model economy, a tax as
high as $20\%$ accelerates the renewable technology adoption by about eleven
years, while a more realistic $2\%$ tax accelerates the transition by only
five years. The tax has the additional effect of a less intensive fossil
fuel use. However, the distortion created by the tax creates a wedge between
the equilibrium and the socially optimal level of investment. Hence, welfare
in the model-economy declines in the tax size.

In our model, subsidies on renewable energy investment also accelerate the rate
of adoption of the renewable energy technology. Indeed, a renewable energy
subsidy appears to be more effective than a tax on fossil fuels, with a $2\%$
subsidy accelerating the introduction of the renewable energy regime by
nineteen years. As a result of the renewable energy subsidy, the fossil fuel
reserves are used more intensively in the short run. This somewhat
paradoxical conclusion can be explained as follows. Since the adoption of
renewable fuel is accelerated as a result of the subsidy, the opportunity
cost of using fossil fuel in the short run declines. Thus, while the subsidy
on renewables leads to a faster transition away from fossil fuels, it also
implies a more intensive use of fossil fuel than what is socially optimal in
the short run. While we do not  explicitly model emissions associated with fossil fuel combustion in our
analysis, it is worth mentioning that this could imply an increase in
negative externalities associated with such emissions in the short run.

Our analysis can be extended in many ways. Introducing technology specific
capital could allow us to more accurately capture the trade-off between
fossil versus renewable energy production. Separating out the effects of learning by doing and explicit investment in R\&D
would allow us to capture innovation and cost reduction in the supply of
renewable energy in greater generality. Studying decentralized allocations
will permit us to explicitly account for creative destruction and the
possibility of under-investment in R\&D. Finally, our current calibration
could be modified to target the economy's initial capital stock. This will
allow us to perform more meaningful welfare comparisons across different
policy regimes. We leave these issues to future research. We believe that
our main findings will remain qualitatively true under such extensions,
which we leave to future research.

\newpage

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\end{document}
